Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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Every elementary matrix is invertible, with inverse given by the reverse elementary operation

Statement

Every elementary matrix E∈Mn(F)E\in M_n(F) is invertible. Its inverse is the elementary matrix belonging to the inverse row operation.

Facts & Assumptions

Given: An elementary matrix EE corresponding to a row operation ρ\rho.

[L1]

The operation ρ\rho has an elementary inverse ρ−1\rho^{-1} (Every elementary row operation has an elementary inverse, so row equivalence is an equivalence relation).

[L2]

Applying an elementary row operation is left multiplication by its elementary matrix (Applying an elementary row operation is left multiplication by its elementary matrix).

[L3]

A square matrix is invertible when it has a two-sided inverse (Invertible matrices and the general linear group GL⁡n(F)\operatorname{GL}_n(F)).

Proof

technique · direct
1.1

Let E′E' be the elementary matrix of ρ−1\rho^{-1}. Applying ρ\rho and then ρ−1\rho^{-1} to InI_n gives E′E=InE'E=I_n, while applying them in the reverse order gives EE′=InEE'=I_n.

L1L2
2.1

Thus E′E' is a two-sided inverse of EE, so EE is invertible and E−1=E′E^{-1}=E'.

step 1.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources